topology of nonarchimedean analytic spaces and relations...

TOPOLOGY OF NONARCHIMEDEAN ANALYTICSPACES AND RELATIONS TO COMPLEX ALGEBRAIC

GEOMETRY

SAM PAYNE

Abstract. This note surveys basic topological properties of nonar-chimedean analytic spaces, in the sense of Berkovich, including therecent tameness results of Hrushovski and Loeser. We also discussinteractions between the topology of nonarchimedean analytic spacesand classical algebraic geometry.

Contents

1. Introduction 12. Nonarchimedean analytification 53. Examples: affine line, algebraic curves, and affine plane 94. Tameness of analytifications 135. Relations to complex algebraic geometry 18References 27

1. Introduction

1.1. Complex algebraic geometry. At its most basic, classical com-plex algebraic geometry studies the common zeros in Cn of a collection ofpolynomials in C[x1, . . . , xn]. Such an algebraic set may have interestingtopology, but is not pathological. It can be triangulated and admits adeformation retract onto a finite simplicial complex. Furthermore, it con-tains an everywhere dense open subset that is a complex manifold, andwhose complement is an algebraic set of smaller dimension. Proceedinginductively, every algebraic set in Cn decomposes as a finite union of com-plex manifolds, and many of the deepest and most fundamental results incomplex algebraic geometry are proved using holomorphic functions anddifferential forms, Hodge theory, and Morse theory on these manifolds.

Partially supported by NSF DMS-1068689 and NSF CAREER DMS-1149054.1

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1.2. Beyond the complex numbers. Modern algebraic geometers areequally interested in the common zeros in Kn of a collection of poly-nomials in K[x1, . . . , xn], for fields K other than the complex numbers.For instance, the field of rational numbers Q is interesting for arith-metic purposes, while the field of formal Laurent series C((t)) is used tostudy deformations of complex varieties. Like C, such fields have naturalnorms. For a prime number p, the p-adic norm | |p is given by writing

a nonzero rational number uniquely as pars

, with a an integer, and p, r,and s relatively prime, and then setting

∣∣∣∣par

s

∣∣∣∣p

= p−a.

The t-adic norm | |t is given similarly, by writing a formal Laurent seriesuniquely as ta times a formal power series with nonzero constant term,and then setting

∣∣∣ta∑

aiti∣∣∣t

= e−a.

One can make sense of convergent power series with respect to thesenorms, and it is tempting to work naively in this context, with “an-alytic” functions given locally by convergent power series. Difficultiesarise immediately, however, for essentially topological reasons, unless thefield happens to be C. The pleasant properties of analytic functions incomplex geometry depend essentially on C being an archimedean field.

1.3. What is an archimedean field? The archimedean axiom saysthat, for any x ∈ K∗, there is a positive integer n such that |nx| > 1.An archimedean field is one in which this axiom holds, such as the realnumbers and the complex numbers. However, there are essentially noother examples. The archimedean axiom is satisfied only by C, withpowers of the usual norm, and restrictions of these norms to subfields.In particular, the only complete archimedean fields are R and C.

1.4. A nonarchimedean field is any other complete normed field.We are not talking about the snake house or a rare collection of exoticcreatures. Nonarchimedean fields are basically the whole zoo. Examplesinclude the completion Qp of Q with respect to the p-adic norm, and thefield of formal Laurent series C((t)). Also, every field is complete andhence nonarchimedean with respect to the trivial norm, given by |x| = 1for x ∈ K∗.

TOPOLOGY OF NONARCHIMEDEAN ANALYTIC SPACES 3

The norm on a nonarchimedean field extends uniquely to its algebraicclosure. The algebraic closure may not be complete1, but the comple-tion of a normed algebraically closed field is again algebraically closed[BGR84, Proposition 3.4.1.3]. So the completion of the algebraic clo-sure of a nonarchimedean field is both nonarchimedean and algebraicallyclosed. A typical example is Cp, the completion of the algebraic closureof Qp.

1.5. The ultrametric inequality. In a nonarchimedean field, a muchstronger version of the triangle inequality holds. The ultrametric inequal-ity says that

|x+ y| ≤ max{|x|, |y|}, with equality if |x| 6= |y|.This property deserves a few moments of contemplation. It implies that,if y is a point in the open ball

B(x, r) = {y ∈ K | |y − x| < r},then B(x, r) = B(y, r). In other words, every point in a nonarchimedeanball is a center of the ball.

Remark 1. Some authors define a nonarchimedean field to be any normedfield that satisfies the ultrametric inequality and do not require thatthe field be complete with respect to the norm. Since completeness isessential for analytic geometry, we maintain this additional hypothesisthroughout.

1.6. Nonarchimedean fields are totally disconnected. Because ofthe ultrametric inequality, the open ball B(x, r) in a nonarchimedeanfield is closed in the metric topology. Since these sets form a basis forthe topology, the field K is totally disconnected. Doing naive analysis onsuch a totally disconnected set is unreasonable. For instance, if f and gare any two polynomials, then the piecewise defined function

Φ(x) =

{f(x) if x ∈ B(0, 1);g(x) otherwise,

is continuous. Even worse, this function Φ is “analytic” in the naivesense that it is given by a convergent power series in a neighborhood ofevery point.

1This is not difficult to see in examples. For instance, the algebraic closure ofC((t)) is the field of Puiseux series C{{t}} =

⋃n C((t1/n)). The exponents appearing

in a Puiseux series have denominators bounded above, but these bounds need not

be uniform on a Cauchy sequence. For instance, the sequence xn =∑n

j=1 tj+ 1

j is

Cauchy, but has no limit in C{{t}}.

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1.7. Grothendieck topologies. Let K be a nonarchimedean field. Theaffine space Kn is totally disconnected in its metric topology, so a purelynaive approach to analytic geometry over K is doomed to fail. Onekludge is to discard the naive notion of topology.

Let us return for a moment to the space of rational numbers Q, which istotally disconnected in its metric topology. The interval [0, 1] in Q is to-tally disconnected and non compact. However, this totally disconnectedset cannot be decomposed into a disjoint union of two open segments withrational centers and rational endpoints. This suggests that [0, 1]∩Q is insome sense connected with respect to finite covers by rational intervals(even though it does decompose as a disjoint union of two open sets, eachof which is an infinite union of rational intervals). This suggests that oneshould restrict to considering finite covers by rational intervals (or atleast covers with a finite refinement) in order to do analysis on Q. Anapproach like this can work, once one gives up the idea that an arbitraryunion of open sets should be open. Naive topology involving open setsand covers by open sets is then replaced by a Grothendieck topology, con-sisting of a collection of covers satisfying certain axioms that are satisfiedby the usual open covers in topology.

1.8. Rigid analytic geometry. John Tate developed a satisfying andpowerful theory of nonarchimedean analytic geometry, based on sheavesof analytic functions in a Grothendieck topology on Kn, when K is al-gebraically closed.2 His theory with this Grothendieck topology is calledrigid analytic geometry. The name “rigid” contrasts these spaces from thenaıve totally disconnected analytic spaces, which Tate called “wobbly.”The fundamental algebraic object in the theory, the ring of convergentpower series on the unit disc, is called the Tate algebra. Algebraic prop-erties of the Tate algebra, including the fact that it is noetherian, play anessential role in all forms of nonarchimedean analytic geometry, whetherone works in the rigid setting or follows the approach of Berkovich. See[BGR84] for a comprehensive treatment of the foundations of rigid ana-lytic geometry.

1.9. Filling in gaps between points. As mentioned above, one candevelop a version of analysis on Q by replacing the metric topology witha suitable Grothendieck topology. Nevertheless, most mathematiciansprefer to add in new points that “fill in the gaps” between the rationalnumbers and do analysis on the real numbers instead. Note the funda-mental absurdity of this construction. Although Q is dense in R, it has

2When K is not necessary algebraically closed, Tate’s theory uses a Grothendiecktopology on K

n/Gal(K|K). Experts will note that this is the set of closed points in

the scheme SpecK[x1, . . . , xn].


measure zero. Once we have filled in the gaps, we can more or less ignorethe points in Q when we do analysis. What is added is so much largerthan what we started with.

In the late 1980s and early 1990s, Vladimir Berkovich developed a newversion of analytic geometry over nonarchimedean fields. At the heartof his construction is a topological space that fills in the gaps betweenthe points of Kn, producing a path connected, locally compact Hausdorffspace that contains Kn with its metric topology, and Kn is dense if thenorm is nontrivial and K is algebraically closed.3 The underlying alge-bra and analysis in Berkovich’s theory are essentially the same as in rigidanalytic geometry, but the topological space is different. The subject ofthis note is the topology of the spaces appearing in Berkovich’s theory,recent results on the tameness of these spaces, and relations betweentopological invariants of these spaces and more classical notions in com-plex algebraic geometry. The first four sections are written for a generalaudience, while the final section, on relations to complex algebraic geom-etry, assumes some familiarity with the cohomology of algebraic varietiesand contains an example illustrating the failure of Lefschetz theoremswith integer coefficients on nonarchimedean analytic spaces.

2. Nonarchimedean analytification

Nonarchimedean analytification is a functor from algebraic varieties(or, more generally, separated schemes of finite type) over a nonar-chimedean field to analytic spaces in the sense of Berkovich. For simplic-ity, we focus on the case of an affine variety. Analytifications of arbitraryvarieties are obtained by a natural gluing procedure from analytifica-tions of affine varieties, which can be described as spaces of seminormson coordinate rings.

2.1. Seminorms on coordinate rings. Let K be a nonarchimedeanfield. Consider polynomials f1, . . . , fr ∈ K[x1, . . . , xn], and let X be thespace of solutions to the corresponding system of equations.4 If x =(x1, . . . , xn) is a point in X(K), then there is an associated seminorm on

3If the norm is nontrivial but K is not algebraically closed, then Kn may notbe dense. However, Berkovich’s analytification also contains K

n/Gal(K|K) with its

natural topology induced by the metric on K, and this subspace is dense.4In other words, X is the Zariski spectrum SpecK[x1, . . . , xn]/(f1, . . . , fr), a locally

ringed space whose underlying topological space is the set of prime ideals p in thisquotient ring, with the Zariski topology. Yoneda’s Lemma identifies this space withthe functor that associates to a K-algebra S the set of solutions to f1, . . . , fr in Sn.In particular, if L|K is an extension field then X(L) is the set of points y ∈ Ln suchthat fi(y) = 0, for 1 ≤ i ≤ r.

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the quotient ring

K[X] = K[x1, . . . , xn]/(f1, . . . , fr).

Here, a seminorm on a ring is simply a function | | from K[X] to R≥0

that satisfies most of the usual axioms of a norm on a field, specificallythat

|fg| = |f | · |g|and

|f + g| ≤ |f |+ |g|.The seminorm | |x associated to a point x in X(K) is given simply by

|f |x = |f(x)|.We will only consider seminorms with the additional property that therestriction to K is the given norm. The given norm on K is nonar-chimedean, and it follows that any seminorm on K[X] extending thisnorm also satisfies the ultrametric inequality

|f + g| ≤ max{|f |, |g|}, with equality if |f | 6= |g|.The one significant difference between norms on fields and seminorms onrings is that a seminorm may take the value zero on a nonzero elementof the ring.

2.2. Analytification of affine varieties. We now describe the ana-lytification of the affine variety X over K in terms of seminorms on itscoordinate ring.

Definition 2. The analytification Xan is the space of all seminorms onthe ring K[X] that extend the given norm on K.5 6

We write x for a point of Xan, when thinking geometrically, and | |xfor the corresponding seminorm on K[X]. The topology on Xan is thesubspace topology for the natural inclusion

Xan ⊂ (R≥0)K[X].

5A word of caution is in order. The system of polynomials f1, . . . , fr may not haveany solutions defined over K. Nevertheless, if K[X] is nonzero then Xan is not empty.One way to see this is to observe that the system f1, . . . , fr has solutions over thealgebraic closure K. Since K is complete, its norm extends uniquely to K, and hencea solution with coordinates in K also determines a point of Xan. More generally, ifL|K is an extension field with a norm that extends the given one on K, then anysolution to f1, . . . , fr with coordinates in L determines a point of Xan.

6An analogous definition over the complex numbers with its archimedean normrecovers classical complex analytic spaces. If X is an affine variety over C then theassociated complex analytic space is naturally identified with the space of seminormson C[X] whose restriction to C is the usual archimedean norm. The bijection takes apoint x ∈ X(C) to the seminorm |f |x = |f(x)|.


This is the coarsest topology such that, for each f ∈ K[X], the functionon Xan given by x 7→ |f |x is continuous.

Theorem 3 ([Ber90]). The topological space Xan is Hausdorff, locallycompact, and locally path connected. The induced topology on the subsetX(K) of points with coordinates in K is the metric topology, and if K isalgebraically closed with nontrivial valuation, then this subset is dense.

In this sense, Xan is a reasonable topological space on which to do analysisthat “fills in the gaps” between the points in the totally disconnected setX(K).

The remainder of Section 2 addresses some of the richer structureson nonarchimedean analytic spaces, beyond the underlying topologicalspace. The casual reader may safely skip ahead to the examples in Sec-tion 3.

2.3. Structure sheaf and morphisms. As an analytic space, Xan

comes with much more structure than just a topology. It has a sheafof analytic functions given locally near each point by limits of rationalfunctions that are regular at that point, and analytic spaces are objectsin a category whose arrows are continuous maps such that the pull backof an analytic function under an analytic map is analytic. There are fur-thermore well-behaved notions of open and closed embeddings, as well asflat, smooth, proper, finite, and etale morphisms in the category of ana-lytic spaces. One example of an etale morphism is given by pulling backthe structure sheaf to a topological covering space, so the fundamentalgroup of the underlying topological space gives essential information re-garding the etale homotopy type of the analytic space. The details maybe found in [Ber93]. Here, we are content simply to study the topologicalspace underlying Xan.

2.4. Projection to the scheme. For those familiar with scheme theoryor Zariski spectra of rings, we explain how Xan relates to SpecK[X], theset of prime ideals in K[X] with its Zariski topology.

Let x be a point in Xan. The set of functions f ∈ K[X] such that|f |x = 0 is a prime ideal p. The seminorm | |x factors through a normon the residue field κp, the fraction field of the quotient K[X]/p, whoserestriction to K is the given one.

Definition 4. For any extension field L|K, let VR(L) be the space of allnorms on L that extend the given norm on K.

The map taking a point in the analytification to the kernel of thecorresponding seminorm gives a natural surjection

Xan → SpecK[X]

8 SAM PAYNE

whose fiber over a point p is VR(κp). In particular, the analytificationdecomposes as a disjoint union

(1) Xan =⊔

p∈XVR(κp).

Note that the topology on the scheme SpecK[X] is never Hausdorff un-less X has dimension zero. Its nonclosed points are the nonmaximalprime ideals p ⊂ K[X], and the closure of p is the irreducible varietySpecK[X]/p. The process of analytification produces a Hausdorff spaceby replacing each nonclosed point p with the space of norms VR(κp). Eachclosed point of SpecK[X] is a maximal ideal m ⊂ K[X]. The residuefield κm at a maximal ideal is algebraic over K [AM69, Proposition 7.9],so the norm on K extends uniquely. Therefore, there is a single point ofXan over each closed point of SpecK[X].

2.5. A quotient description of Xan. The decomposition above showsthat each point of Xan is associated to a point of SpecK[X] togetherwith a norm on its residue field that extends the given one on K. Forsome purposes, rather than keeping track of all of these residue fields,it makes more sense to consider points defined over arbitrary normedextensions L|K. One can still recover the analytification Xan by takinga quotient by an appropriate equivalence relation, as described below.The Zariski spectrum SpecK[X] has an analogous description, in termsof natural equivalence classes of points over extension fields of K. Thekey difference here is the role played by norms.

A normed extension L|K is a field L together with a norm | | : L→ R≥0

that extends the given norm on K. We consider triples consisting of anextension field of K, a norm that extends the given one, and a point ofX over this normed extension, and the equivalence relation generated bysetting

(L, | |, x) ∼ (L′, | |′, x′)whenever there is an embedding L ⊂ L′ such that the restriction of | |′ toL is | | and x is identified with x′ by the induced inclusion X(L) ⊂ X(L′).

Proposition 5. The analytification Xan is the space of equivalence classesof points of X over normed extensions of K:

Xan = {(L, | |, x)}/ ∼ .

Much of the recent progress in understanding the topology of nonar-chimedean analytic spaces has come through logic and model theory,and this description of Xan in terms of equivalence classes of points overnormed extensions is closest in spirit to the spaces of stably dominatedtypes that appear prominently in this context. Note, however, that model


theorists typically consider seminorms into ordered groups of arbitraryrank, not only the real numbers.

3. Examples: affine line, algebraic curves, and affineplane

IfX has dimension 0 thenXan is equal toX, and both have the discretetopology. We now consider the first nontrivial cases of analytifications.

3.1. Analytification of the line: trivial norm. The simplest exampleto consider is the affine line

A1 = SpecK[y],

in the case where the norm on K is trivial. Let x be a point in (A1)an.If |y|x is greater than 1 and

f = a0 + a1y + · · ·+ adyd

is a polynomial of degree d, then |f |x = |y|dx. Therefore, | |x is uniquelydetermined by |y|x, and the limit, as |y|x goes to 1, is the trivial normη on the function field K(y). This gives an embedded copy of [1,∞) in(A1)an.

Now, suppose | |x is not trivial, and |y|x ≤ 1. Then | |x is less thanor equal to 1 on all of K[y], and the set of f such that |f |x < 1 is anonzero prime ideal. Each such ideal is generated by a unique irreduciblemonic polynomial g ∈ K[y]. Given such a g and a real number t < 1,there is a unique seminorm on K[y] such that |g| = t; it takes ga · hto at, for h relatively prime to g. The limit of these seminorms as tgoes to 1 is again the trivial norm η, and the limit as t goes to 0 isthe closed point corresponding to the maximal ideal mg generated by g.This gives a rough picture of (A1)an as a sort of tree, with an infinitestem consisting of seminorms on K(y) that are greater than 1 on y,and infinitely many branches that end in leaves corresponding to theirreducible polynomials in K[y]. Equivalently, the leaves correspond toclosed points in the scheme A1 over K, or elements of K/Gal(K|K).

Some discussion of the topology on this tree is in order. The topol-ogy on the subset where |y| ≤ 1 is not the cone over the discrete setK/Gal(K|K). Rather, it is an inverse limit of cones over finite subsetsof K/Gal(K|K), so any neighborhood of η contains all but finitely manyof the branches. To see this, note that for any f ∈ K[y], the inducedfunction (A1)an → R ∪∞ taking x to |f |x is 1 on almost all of the rays,all except those corresponding to irreducible factors of f . Therefore, thepreimage of any neighborhood of 1 in R≥0 contains all but finitely manyof the branches, and these form a basis for the neighborhoods of η.

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is a polynomial of degree d, then valx(f) = dvalx(y). Therefore, valx isuniquely determined by valx(y), and the limit, as valx(y) goes to 0, is thetrivial valuation ⌘ on the function field K(y). This gives an embeddedcopy of R0 in (A1)an.

Now, suppose valx is not trivial, and valx(y) � 0. Then valx is nonneg-ative on all of K[y], and the set of f such that valx(f) > 0 is a nonzeroprime ideal. Each such ideal is generated by a unique irreducible monicpolynomial g 2 K[y]. Given such a g and a positive real number t, thereis a unique valuation on K[y] such that val(g) = t; it takes ga · h toat, for h relatively prime to g. The limit of these valuations as t goesto 0 is again the trivial valuation ⌘, and the limit as t goes to 1 is theclosed point corresponding to the maximal ideal mg generated by g. Thisgives a rough picture of (A1)an as a sort of tree, with an infinite stemconsisting of valuations on K(y) that are negative on y, and infinitelymany branches leading to leaves corresponding to the irreducible poly-nomials in K[y]. Equivalently, the leaves correspond to closed points inthe scheme A1 over K, or elements of K/Gal(K|K).

Figure 1. The analytification of A1 with respect to thetrivial valuation.

Some discussion of the topology on this tree is in order. The topologyon the subset where val(y) � 0 is not the cone over the discrete setK/Gal(K|K). Rather, it is an inverse limit of cones over finite subsetsof K/Gal(K|K), so any neighborhood of ⌘ contains all but finitely manyof the branches. To see this, note that for any f 2 K[y], the inducedfunction (A1)an ! R[1 taking x to valx(f) is zero on almost all of therays, all except those corresponding to irreducible factors of f . Therefore,the preimage of any neighborhoods of 0 in R[1 contains all but finitelymany of the branches, and these form a basis for the neighborhoods of⌘.

Figure 1. The analytification of A1 with respect to thetrivial norm.

In this way, not only does VR(K(y)) fill in the gaps to connect the setof closed points in A1

K with the discrete (metric) topology, it also inter-polates between the metric topology and the cofinite (Zariski) topologyin a subtle way.

3.2. Analytification of the line: nontrivial norms. The analytifi-cation of A1 in the case of a nontrivial norm is again a tree, but now theset of branch points is dense. At each branch point, the local topologyis like the topology at η in the analytification of the line with respect tothe trivial norm. It is described beautifully, and in detail, in Section 1 of[Bak08a].

3.3. Analytification of curves. The analytification of an arbitrarysmooth curve X looks locally similar to that of the line. If the normis trivial, then Xan has finitely many open branches, corresponding tothe points of the smooth projective model that are not in X, and therest is an inverse limit of cones over finite subsets of X(K)/Gal(K|K).

If the norm on K is nontrivial, then Xan is locally homeomorphicto (A1)an, but may have nontrivial global topology, as in the followingexample.

The dual graph of the special fiber of a semistable formal model embedsin Xan as a strong deformation retract. For instance, an elliptic curvewith bad reduction has a semistable formal model whose special fiber isa loop of copies of P1, and its analytification deformation retracts ontoa circle. Every finite graph occurs in this way, as the dual graph of thespecial fiber of a formal model, and hence as a deformation retraction ofan analytic curve.


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In this way, not only does VR(K(y)) fill in the gaps to connect the setof closed points in A1

K with the discrete (metric) topology, it also inter-polates between the metric topology and the cofinite (Zariski) topologyin a subtle and beautiful way.

3.2. Analytification of the line: nontrivial valuation. The analyti-fication of A1 in the case of a nontrivial valuation is again a tree, butnow the set of branch points is dense. At each branch point, the localtopology is like the topology at ⌘ in the analytification of the line withrespect to the trivial valuation. It is described beautifully, and in detail,in Section 1 of [Bak08a].

3.3. Analytification of curves. The analytification of an arbitrarysmooth curve X looks locally similar to that of the line. If the valu-ation is trivial, then Xan has finitely many open branches, correspondingto the points of the smooth projective model that are not in X, and therest is an inverse limit of cones over finite subsets of X(K)/Gal(K|K).

If the valuation on K is nontrivial, then Xan is locally homeomomor-phic to (A1)an, but may have nontrivial global topology, as in the follow-ing example.

Figure 2. The analytification of a smooth curve with re-spect to a nontrivial valuation.

The dual graph of the special fiber of a semistable formal model embedsin Xan as a strong deformation retract. For instance, an elliptic curvewith bad reduction has a semistable formal model whose special fiber isa loop of copies of P1, and its analytification deformation retracts ontoa circle. Every finite graph occurs in this way, as the dual graph of thespecial fiber of a formal model, and hence as a deformation retraction ofan analytic curve.

Figure 2. The analytification of a smooth curve with re-spect to a nontrivial norm.

See [BPR11, Section 5] for further details on the structure theory ofnonarchimedean analytic curves in the case where K is algebraicallyclosed. The general case is similar; if K is not algebraically closed thenXan is the analytification of the base change to the completion of thealgebraic closure, modulo the action of Gal(K|K).

In the case where K has a countable dense subset, as is the case for

Qp, Cp, and Fp((t)), the topology of analytic curves over K is locallymodeled on that of the “universal dendrite,” each such curve admits adeformation retract onto a finite graph. If the graph is planar then thecurve admits an embedding in the euclidean plane R2, and if the graphis not planar then the curve embeds in R3 [HLP12].

3.4. Toward the analytification of the affine plane. Let us try toform a mental image of the analytification of the affine plane, using thediscussion of curves, above, and the decomposition (1), in the case wherethe norm is trivial. For another approach to visualizing the local topologyof (A2)an, with illustrations, see [Jon12, Section 6.7].

To start, note that (A2)an contains the analytification of any planecurve, and the complement of the union of these analytic curves is thespace VR(K(x1, x2)) of real norms on the function field K(x1, x2) thatare trivial on K. Just as the closed points of a curve X lie at the endsof infinite branches of VR(K(X)), the analytifications of curves in A2

lie in some sense at infinity, as limits of two-dimensional membranes inVR(K(x1, x2)). Of course, the situation is somewhat more complicated,since the analytifications of distinct curves are joined at their points ofintersection.

So, imagine a network of analytic curves at infinity, one for each curvein the plane, glued along leaves of the infinite branches corresponding to

12 SAM PAYNE

their points of intersection. We now begin to describe how VR(K(x1, x2))fills in the interior of this network. Suppose Y and Z are curves in A2

that meet transversely at a point x. Let f and g be defining equationsfor Y and Z, respectively. These can be interpreted as local coordinateson A2 at x, so any function h ∈ K[x1, x2] can be expanded locally nearx as a power series in f and g.

There is then a cone of “monomial norms” in these local coordinates,for which the norm of a function depends only on the monomials in fand g that appear in its power series expansion, and these norms aredefined as follows. Let v = (v1, v2) be a point in the cone R2

≥0, and let hbe a function whose local power series expansion near x is

h =∑

aijfigj.

Then the norm corresponding to v takes h to

|h|v = maxi,j{e−iv1−jv2},

where the maximum is taken over pairs (i, j) such that aij 6= 0.The closure of this cone in (A2)an is a copy of (R≥0 ∪∞)2, joining the

trivial norm η on K(x1, x2) to the trivial norms ηY and ηZ on K(Y ) andK(Z), respectively, and the point x.12 SAM PAYNE

⌘

x⌘Y

⌘Z

Figure 3. The closure of a cone of monomial valuations in (A2)an.

In the geometry of this cone, the limit of any ray with positive finite slopeis x, so it is perhaps best imagined as a curved membrane stretching aninfinite distance toward x, from the frame formed by the rays joining ⌘X

and ⌘X0 .Understanding how all of these membranes fit together in (A2)an is

challenging, especially as one must also keep track of the topology ina neighborhood of ⌘. Moreover, we are still far from a full descriptionof the underlying set of (A2)an. All of the valuations on K(y, z) thatare monomial in some system of local coordinates are of the simplestflavor in the sense of pure valuation theory; they satisfy Abhyankar’sinequality6 with equality. There are many valuations on K(y, z) thatare not monomial in any system of coordinates. For instance, a pairof formal power series in K[[t]] define a formal germ of a curve in A2,and there is a valuation on K[y, z] obtained by pulling functions backto this germ and computing order of vanishing at t = 0. If these powerseries are algebraically independent over K, then the image of this germis Zariski dense in A2, and Abhyankar’s inequality is strict. These pointsare outside of the infinite union of membranes described above.

Each valuation corresponding to a point in (A2)an, including thosewhere Abhyankar’s inequality is strict, may be obtained as a limit ofmonomial valuations in various systems of local coordinates (or even asa limit of valuations corresponding to closed points), and the same is truein higher dimensions and on singular spaces, but the precise way that allof the pieces fit together becomes more and more di�cult to describe.

6Recall that Abhyankar’s inequality says that transcendence degree of the residuefield extension plus the rank of the extension of the value group is less than or equalto the transcendence degree of the total extension [Abh56].

Figure 3. The closure of a cone of monomial norms in (A2)an.

In the geometry of this cone, the limit of any ray with positive finite slopeis x, so it is perhaps best imagined as a curved membrane stretching aninfinite distance toward x, from the frame formed by the rays joining ηYand ηZ .

Understanding how all of these membranes fit together in (A2)an ischallenging, especially as one must also keep track of the topology in


a neighborhood of η. Moreover, we are still far from a full descriptionof the underlying set of (A2)an. All of the norms on K(x1, x2) that aremonomial in some system of local coordinates are of the simplest flavorin the sense of pure normed field theory; they satisfy Abhyankar’s in-equality7 with equality. There are many norms on K(x1, x2) that are notmonomial in any system of coordinates. For instance, a pair of formalpower series in KJtK define a formal germ of a curve in A2, and there isa seminorm on K[x1, x2] obtained by pulling functions back to this germand exponentiating minus the order of vanishing at t = 0. If these powerseries are algebraically independent over K, then the image of this germis Zariski dense in A2, and Abhyankar’s inequality is strict. These pointsare outside of the infinite union of membranes described above.

Each seminorm corresponding to a point in (A2)an, including thosewhere Abhyankar’s inequality is strict, may be obtained as a limit ofmonomial seminorms in various systems of local coordinates (or even asa limit of seminorms corresponding to closed points), and the same istrue in higher dimensions and on singular spaces, but the precise waythat all of the pieces fit together becomes more and more difficult todescribe.

4. Tameness of analytifications

Beyond the case of curves, which can be treated more or less by hand,it is not obvious that analytifications of algebraic varieties are not patho-logical topological spaces. In his deep work on skeletons of formal models[Ber99, Ber04], Berkovich proved that analytifications of smooth varietiesare locally contractible and have the homotopy type of a finite simplicialcomplex, provided that the norm is nontrivial. However, the correspond-ing statements for singular varieties in positive and mixed characteristic,and for varieties over trivially normed fields, have so far eluded proofby such methods. For instance, while it has been known for some timethat the analytification of a smooth variety over a trivially normed fieldis contractible, the only known proof that it is locally contractible isquite recent and passes through model theory and spaces of stably dom-inated types. This local contractibility is just one of the fundamentalconsequences of the tameness theorem of Hrushovski and Loeser [HL12].

4.1. Analytic domains. We now move toward describing the local struc-ture of analytic spaces. In algebraic geometry, affine open subvarieties arethe basic building blocks. An arbitrary algebraic variety is constructedby gluing its affine open subvarieties along open immersions, and each

7Recall that Abhyankar’s inequality says that transcendence degree of the residuefield extension plus the rank of the group extension given by the images of the normsis less than or equal to the transcendence degree of the total extension [Abh56].

14 SAM PAYNE

affine variety is determined by its ring of global regular functions. Affi-noid analytic domains play a similar role as basic building blocks innonarchimedean analytic geometry. An arbitrary nonarchimean analyticvariety is constructed by gluing its affinoid analytic domains along im-mersions, and each affinoid domain is determined by its coordinate ringof global analytic functions.8

We are particularly interested in analytifications of algebraic varieties,and now describe the affinoid domains in these spaces. Let X be an affinevariety. A typical affinoid domain in Xan can be realized by choosing aclosed embedding ι : X ↪→ An and intersecting with the unit ball. Moreprecisely, if

ι∗ : K[y1, . . . , yn]→ K[X]

is the corresponding surjection of K-algebras, then the subset

U = {x ∈ Xan | |ι∗(yi)|x ≤ 1, for 1 ≤ i ≤ n}is an affinoid analytic domain. As the terminology suggests, the affinoidanalytic domain U inherits the structure of a nonarchimedean analyticspace from its inclusion in Xan.

The role of more general, not necessarily affine, open subvarieties inalgebraic geometry is played by compact analytic domains in nonar-chimedean analytic geometry. The compact analytic domains are exactlythe finite unions of affinoid analytic domains. Every point in an analyticspace has a basis of neighborhoods consisting of compact analytic do-mains, so understanding the topology of compact analytic domains isessential to understanding the local topology of analytic spaces.

4.2. Skeletons of formal models. Assume the norm on K is nontriv-ial. Then any compact analytic domain U ⊂ X has formal models, whichmeans that it can be realized as the “analytic generic fiber” of a formalscheme. Roughly speaking, a formal model of U is given by a compatiblesystem of local coordinates on X such that the points in U are exactlythose whose local coordinates are in the valuation ring R ⊂ K. In theselocal coordinates, one can then mod out by the maximal ideal m ⊂ Rto get the special fiber of the formal model, which is a scheme over theresidue field k = R/m.

Example 6. Fix an affine embedding X ⊂ SpecK[x1, . . . , xn]. Then wecan “clear denominators” in the coordinate ring K[x1, . . . , xn]/IX to get

8Affine varieties and affinoid domains also share important cohomological and sheaftheoretic properties. Each coherent sheaf on an affinoid domain is associated to afinitely generated module over its coordinate ring and has vanishing higher cohomol-ogy [BGR84, 8.2.1 and 9.4.3]. For the corresponding properties of affine varieties, see[Har77, II.5.1 and III.3.5].


a finitely presented R-algebra

A = R[x1, . . . , xn]/(IX ∩R[x1, . . . , xn]).

The scheme SpecA is an integral model of X. Its generic fiber is X andits special fiber is SpecA ⊗R k. For any nonzero $ ∈ m, the $-adic

completion of this integral model is a formal model Spf A with analyticgeneric fiber

U = {(x1, . . . , xn) ∈ Xan | |xi| ≤ 1 for 1 ≤ i ≤ n}.The special fiber of Spf A is the same as the special fiber of SpecA.

Remark 7. Just as one can clear denominators on coordinate rings toget formal models of compact analytic domains, one can also clear de-nominators on morphisms to get morphisms between such formal models.Raynaud famously proved that the category of quasi-compact and quasi-separated rigid analytic spaces is naturally identified with a localizationof the category of quasi-compact formal schemes, in which admissible for-mal blowups (modifications of formal schemes that affect only the specialfiber) are inverted. See also [Ray74, BL93a, BL93b, BLR95a, BLR95b]for further details on formal schemes and their relation to analytic ge-ometry.

If a compact analytic domain U has a formal model with an especiallynice special fiber9 then its topology is controlled by the combinatorics ofthe special fiber.

Definition 8. A formal model is strictly semistable if its special fiber isa reduced union of smooth varieties meeting transversally.

The dual complex of the special fiber of a strictly semistable formal modelof U is a regular ∆-complex with one vertex for each irreducible compo-nent, one edge for each irreducible component of a pairwise intersection,one 2-face for each irreducible component of a triple intersection, andso on. It has a canonical realization as a closed subset of U . Roughlyspeaking, the dual complex consists of seminorms that are monomial inthe local coordinates defining the model.

Theorem 9 (Berkovich). If a compact analytic domain has a strictlysemistable formal model, then it admits a strong deformation retract ontothe dual complex of its special fiber. In particular, it has the homotopytype of a finite simplicial complex.

9In residue characteristic zero, formal models with nice special fibers are con-structed using the semistable reduction theorem [KKMSD73], a version of resolutionof singularities for a one-parameter family of varieties. Berkovich also proved localcontractibility of smooth analytic spaces over nontrivially valued fields with positiveresidue characteristic using formal models with nice special fibers [Ber99], which heconstructed via de Jong’s theorem on alterations [dJ96].

16 SAM PAYNE

Berkovich constructed skeletons and deformation retractions much moregenerally, for analytic spaces with polystable formal models, and usedthese skeletons to prove tameness results, including local contractibility,provided that such formal models exist, as is the case in residue character-istic zero [Ber99, Ber04]. See also [Nic11, Section 3 and Proposition 4.4]and [MN12, Section 3] for an accessible treatment of the semistable case.Other recent work of Nicaise and Xu shows that similar results hold foralgebraic formal models whose special fibers have normal crossings butare not necessarily reduced [NX13].

Remark 10. This discussion of skeletons of formal models assumes thatthe norm is nontrivial. See [Thu07] for a closely related construction ofskeletons and deformation retractions associated to toroidal embeddingsin the case where the norm is trivial.

4.3. Semialgebraic sets and tameness. As mentioned above, Ber-kovich proved that skeletons of sufficiently nice formal models control thetopology of analytic spaces when the norm is nontrivial and such formalmodels exist, as is the case when the residue field has characteristic zero.The recent work of Hrushovski and Loeser proves similar tameness resultswith no hypothesis on the normed field, by very different methods. Tostate their tameness theorem, it is most helpful to talk about subsets ofanalytifications that are more general than compact analytic domains.

Definition 11. Let X be an affine algebraic variety over K. A semial-gebraic subset U ⊂ Xan is a finite boolean combination of subsets of theform

{x ∈ Xan | |f |x ./ |g|λx},with f, g ∈ K[X], λ ∈ R, and ./∈ {≤,≥, <,>}.A semialgebraic set is definable if the conditions defining the subset canbe chosen such that some power of λ is in |K|.

By construction, every point in Xan has a basis of neighborhoods thatare semialgebraic sets, and if the norm is nontrivial then these sets can bechosen to be definable. The Gerritzen-Grauert theorem on locally closedimmersions of affinoid varieties [BGR84, Theorem 7.3.5.1] guaranteesthat affinoid domains and, more generally, compact analytic domains inXan are semialgebraic subsets.

We now state a basic version of the main result from [HL12].

Theorem 12. Let U ⊂ Xan be a semialgebraic subset. Then there isa finite simplicial complex ∆ ⊂ U , of dimension less than or equal todim(X), and a strong deformation retraction U × [0, 1]→ ∆.

Since ∆ is locally contractible, and the topology on Xan has a semialge-braic basis, it follows that Xan is locally contractible. The homotopy type


of the complex ∆ is a fundamental invariant of a semialgebraic space U ,and many applications to complex geometry involve understanding thecohomology and fundamental groups of these complexes.

The approach of Hrushovski and Loeser does not involve the construc-tion of nice models or toroidal compactifications. It thereby avoids reso-lution of singularities and is insensitive to the residue characteristic. Asmentioned above, the proof involves a detailed study of spaces of stablydominated types, a notion coming from model theory [HHM08]. The caseof curves is treated by hand, and the general case is a subtle induction ondimension, which involves birationally fibering an n-dimensional varietyby curves over a base of dimension n− 1. In particular, the proof of thetameness theorem for a single variety in dimension n requires a tamenessstatement for families of varieties in lower dimensions. See the Bourbakinotes of Ducros [Duc13] for an excellent introduction to this work, andthe original paper [HL12] for further details.

4.4. Limits of skeletons. The finite simplicial complexes constructedin [HL12], which live inside semialgebraic sets as strong deformation re-tracts, are also called skeletons. Hrushovski and Loeser prove much morethan the existence of a single skeleton. Each semialgebraic set U of pos-itive dimension contains infinitely many skeletons ∆i, with natural pro-jections between them, and the semialgebraic set is recovered as the limitof this inverse system

lim←−∆i = U.

Furthermore, each of these projections has a natural section, and theunion lim−→∆i is the subset of U consisting of points corresponding toAbhyankar seminorms.

Similar constructions involving limits of skeletons of formal modelswere considered earlier by Berkovich and in [KT02, KS06]. See [BPR11,Corollary 5.56 and Theorem 5.57] for an explicit treatment of such limitsfor curves.

4.5. Limits of tropicalizations. The topological space Xan can alsobe realized as a limit of finite polyhedral complexes using tropical ge-ometry [Pay09, FGP12]. In this approach, the polyhedral complexes aretropicalizations of algebraic embeddings of X in toric varieties, with pro-jections induced by toric morphisms that commute with the embeddings.The construction of this inverse system is essentially elementary, at leastin the quasiprojective case, but it does not lead to a proof of tameness.It is not even known whether there exists a single tropicalization suchthat the projection from Xan is a homotopy equivalence, and the relationbetween these tropical inverse systems and the skeletons of Hrushovskiand Loeser remains unclear.

18 SAM PAYNE

5. Relations to complex algebraic geometry

The link between algebraic and analytic geometry over nonarchimedeanfields is as close as the link between complex algebraic and complex an-alytic geometry. Coherent algebraic sheaves have coherent analytifica-tions, analytifications of etale algebraic morphisms are etale, and thereare comparison theorems for `-adic etale cohomology [Ber90, Ber93]. Itis not at all surprising, then, that nonarchimedean analytic techniquesare powerful for studying algebraic varieties over nonarchimedean fields.

However, nonarchimedean analytic techniques are also powerful forstudying algebraic varieties over the complex numbers. One reason issimple: nonarchimedean fields such as Cp and the completion of C{{t}}are isomorphic to C as abstract fields. The isomorphism is not explicitor geometric, but elimination of quantifiers for algebraically closed fieldsimplies that any two uncountable algebraically closed fields of the samecardinality and characteristic are isomorphic [Mar02, Proposition 2.2.5].In particular, whenever one can use nonarchimedean analytic techniquesto produce a variety over Cp with a certain collection of algebraic proper-ties, it follows that there exists a variety over C with the same collectionof properties.

Perhaps more surprisingly, one can also get significant mileage bystudying analytifications of open and singular complex varieties withrespect to the trivial norm and their semialgebraic subsets. See the dis-cussion of Milnor fibers and analytic links, below.

5.1. Tropical linear series. Many applications of nonarchimedean an-alytic spaces in complex geometry involve less information than the fullstructure sheaf, but more than the mere topological space. Tropical ge-ometry resides firmly in this intermediate realm. For instance, if X is acurve then VR(K(X)), the complement in Xan of the set of closed pointsof X, inherits a natural metric. Through the tropical Riemann-Rochtheorem [BN07, GK08, MZ08], Baker’s specialization lemma and its gen-eralizations [Bak08b, AB12, AC13], the nonarchimedean Poincare-Lelongformula [Thu05, BPR11], and the theory of harmonic morphisms of met-ric graphs [BN09, ABBR13], this metric is a powerful tool in the studyof linear series on algebraic curves. It has been used to characterize dualgraphs of special fibers of regular semistable models of curves of a givengonality [Cap12], to compute the gonality of curves that are generic withrespect to their Newton polygon [CC12], to characterize the Newton poly-gons of Brill-Noether general curves in toric surfaces [Smi14], to boundthe gonality of Drinfeld modular curves [CKK12], and to give new proofsof the Brill-Noether and Gieseker-Petri theorems [CDPR12, JP14].


In the remaining sections, we survey some of the applications of nonar-chimedean analytic geometry to classical complex varieties that involveonly the topology of analytifications.

5.2. Singular cohomology. Let X be an algebraic variety over thecomplex numbers, and letX(C) be the associated complex analytic space.Recall that Deligne defined a canonical mixed Hodge structure on the ra-tional cohomology H∗(X,Q), and one part of this structure is the weightfiltration

W0Hk(X(C),Q) ⊂ · · · ⊂ W2kH

k(X(C),Q) = Hk(X(C),Q),

which is strictly functorial for algebraic morphisms. This means that iff : X ′ → X is a morphism then

f ∗(Hk(X(C),Q)) ∩WjHk(X ′(C),Q) = f ∗(WjH

k(X(C),Q)).

If X is smooth and compact then Wk−1Hk(X,Q) = 0 and WkH

k(X,Q) =Hk(X,Q). In other words, Hk(X(C),Q) is of pure weight k. In thegeneral case, where X may be singular and noncompact, the gradedpieces of Hk of weight less than k encode information on the singularitiesof X, while the pieces of weight greater than k encode information aboutthe link of the boundary in a compactification. This is perhaps bestillustrated by an example.

Example 13. Consider a curve X of geometric genus 1, with three punc-tures and a single node x, as shown.

18 SAM PAYNE

In the remaining sections, we survey some of the applications of nonar-chimedean analytic geometry to classical complex varieties that involveonly the toplogy of analytifications.

5.2. Singular cohomology. Let X be an algebraic variety over thecomplex numbers, and let X(C) be the associated complex analytic space.Recall that Deligne defined a canonical mixed Hodge structure on the ra-tional cohomology H⇤(X, Q), and one part of this structure is the weightfiltration

W0Hk(X(C), Q) ⇢ · · · ⇢ W2kH

k(X(C), Q) = Hk(X(C), Q),

which is strictly functorial for algebraic morphisms. This means that iff : X 0 ! X is a morphism then

f ⇤(Hk(X(C), Q) \WjHk(X 0(C), Q) = f ⇤WjH

k(X(C), Q).

If X is smooth and compact then Wk�1Hk(X, Q) = 0 and WkH

k(X, Q) =Hk(X, Q). In other words, Hk(X(C), Q) is of pure weight k. In thegeneral case, where X may be singular and noncompact, the gradedpieces of Hk of weight less than k encode information on the singularitiesof X, while the pieces of weight greater than k encode information aboutthe link of the boundary in a compactification. This is perhaps bestillustrated by an example.

Example 14. Consider a curve X of geometric genus 1, with three punc-tures and a single node x, as shown.

Figure 4. A nodal curve with three punctures

Its normalization eX is obtained by resolving the node. The homologyH1(X(C), Q) is generated by a loop � through the node, two loops around

Figure 4. A nodal curve with three punctures

Its normalization X is obtained by resolving the node. The homologyH1(X(C),Q) is generated by a loop through the node, two loops around

20 SAM PAYNE

the doughnut on the left, and two loops around punctures. In the corre-sponding dual basis for H1(X(C),Q), the class dual to the loop throughthe node generates W0H

1(X(C),Q).We now consider the nonarchimedean analytification of X with re-

spect to the trivial norm on C. As in the examples from Section 3,

the analytification of the normalization X of X is an infinite tree, withthree unbounded branches corresponding to the punctures. The remain-

ing branches end in leaves, corresponding to the closed points of X and,in Xan, two of these closed points are identified at the node x. Theanalytification is therefore as shown here.


the genus, and two loops around the punctures. In the corresponding dualbasis for H1(X(C), Q), the class �⇤ generates W0H

1(X(C), Q).We now consider the nonarchimedean analytification of X with respect

to the trivial valuation on C. As in the examples from Section 3, the

analytification of the normalization eX of X is an infinite tree, with threeunbounded branches corresponding to the punctures. The remaining

branches end in leaves, corresponding to the closed points of eX and,in Xan, two of these closed points are identified at the node x. Theanalytification is therefore as shown here.

Figure 5. The analytification of a nodal curve with three punctures.

Note that the identification of two closed points in eX creates an extraloop in both X(C) and Xan and that, on X(C), the new class in H1 hasweight zero.

Theorem 15 ([Ber00]). Let X be a complex algebraic variety, and letXan be its analytification with respect to the trivial valuation on C. Thenthere is a natural isomorphism

H⇤(Xan, Q) ⇠= W0H⇤(X(C), Q).

A similar result holds for varieties defined over a local field, such asQp. In these cases, Hk(Xan, Q`) is canonically identified with the weightzero part of Hk

et(X, Q`).Singular cohomology of skeletons and their relations to weight filtra-

tions have also appeared in the tropical geometry literature, for instance

Figure 5. The analytification of a nodal curve with three punctures.

Note that the identification of two closed points in X creates an extraloop in both X(C) and Xan and that, on X(C), the new class in H1 hasweight zero.

Theorem 14 ([Ber00]). Let X be a complex algebraic variety, and letXan be its analytification with respect to the trivial norm on C. Thenthere is a natural isomorphism

H∗(Xan,Q) ∼= W0H∗(X(C),Q).


A similar result holds for varieties defined over a local field, such asQp. In these cases, Hk(Xan,Q`) is canonically identified with the weightzero piece of the monodromy filtration on Hk

et(X,Q`).Singular cohomology of skeletons and their relations to weight filtra-

tions have also appeared in the tropical geometry literature, for instancein [Hac08, HK12, KS12a]. The key fact is that tropicalizations are skele-tons in the case where all initial degenerations are smooth and irreducible,and there are natural parametrizing complexes for tropicalizations thatare skeletons more generally, in the “schon” case, where all initial degen-erations are smooth, but possibly reducible. See [Gub13] for details onthe relation between tropicalizations, initial degenerations, and formalmodels.

5.3. Beyond rational cohomology. There is far more information inthe topology of Xan than just its rational cohomology. For instance, atotally degenerate Enriques surface has no `-adic rational cohomologyof weight zero in degree above zero, but its analytification is not con-tractible. It has the homotopy type of RP2. In this case, the fundamentalgroup of Xan agrees with the etale fundamental group of X. The natureof the relationship between the fundamental group and higher homotopygroups of an analytification and the algebraic invariants of the varietyis not known in general. Subtleties appear already when one examinestorsion in the cohomology of analytifications. For instance, the naivegeneralization of the Lefschetz Hyperplane Theorem does not hold forcohomology of nonarchimedean analytic spaces, as explained in the nextsection.

5.4. Lefschetz Hyperplane Theorems. Suppose D is a hyperplanesection of a smooth projective algebraic variety over K. The `-adic Lef-schetz theorem [Del80, Section 4.1.6] says that the natural restrictionmaps

H iet(X,Z`)→ H i

et(D,Z`)are isomorphisms for i < dimD and injective for i = dimD. By tensoringwith Q` and applying the weight zero comparison theorems, the compati-bility of restrictions with weight filtrations, and the universal coefficientstheorem in singular cohomology, it follows that the natural maps

H i(Xan,Q)→ H i(Dan,Q)

are also isomorphisms for i < dimD and injective for i = dimD.Such Lefschetz Theorems also hold in the singular cohomology of com-

plex varieties, and in this classical context they can be extended to in-tegral cohomology and even homotopy groups. Furthermore, the hy-perplane section can be replaced by an arbitrary ample divisor, i.e. adivisor D such that mD is a hyperplane section for some positive integer

22 SAM PAYNE

m. The proof is by generalized Morse theory; a smooth complex varietyof dimension n can be obtained from any ample divisor by adding cellsof dimension at least n [Bot59]. Similar phenomena appear sometimesin tropical and nonarchimedean analytic geometry. See, for instance,[AB14] for Lefschetz hyperplane theorems on locally matroidal tropicalvarieties.

Example 15. The analytification of the Jacobian of a totally degeneratecurve of genus g has a skeleton which is a real torus of dimension g. Thisskeleton is naturally identified with the tropical Jacobian of the skeletonof the curve [BR13]. Furthermore, the image of the analytification ofthe theta divisor is the tropical theta divisor, and the tropical Jacobianis obtained from this tropical theta divisor by attaching a single cell ofdimension g [MZ08]. This suggests that Lefschetz theorems may holdfor the inclusion of the analytic theta divisor in the analytic Jacobianfor integral cohomology and homotopy groups, though it seems to beunknown, in general, whether the projection from the analytification ofthe theta divisor to the tropical theta divisor is a homotopy equivalence.

However, it is not true that the inclusion of the analytification of an ampledivisor in a variety of dimension n induces isomorphisms on integralcohomology groups in degrees up to n − 2, as the following exampleshows.

Example 16. Let E be an elliptic curve over the complex numbers, witha 2-torsion point q. Let E[2] be the 2-torsion subgroup, which we view asa divisor of degree 4 on E. The abelian 3-foldX = E×E×E has an ampledivisor with simple normal crossings D = p∗1(E[2]) ∪ p∗2(E[2]) ∪ p∗3(E[2]).The dual complex ∆(D) has 12 vertices, 48 edges, and 64 2-faces; it issimply connected and homotopy equivalent to a wedge sum of 27 spheres.Note that (q, q, q) acts on X by a fixed point free involution that preservesD, and the induced action on ∆(D) is also fixed point free. The quotient(X ′, D′) is again an abelian 3-fold with an ample divisor with simplenormal crossings, and ∆(D)→ ∆(D′) is a universal cover. In particular,

H1(∆(D′),Z) ∼= π1(∆(D′)) ∼= Z/2Z.Let K be the completion of the algebraic closure of C((t)). We claim

that there is a smooth ample divisor H in X ′K , linearly equivalent toD′, whose analytification is homotopy equivalent to ∆(D′). To provethe claim, we first show that D′ is basepoint free. Note that D′ is thepullback of a divisor D′′ on E/q × E/q × E/q, and D′′ is the union ofthe 3 pullbacks of a divisor of degree 2 on the elliptic curve E/q. Anydivisor of degree 2 on an elliptic curve is basepoint free. It follows thatD′′ and D′ are basepoint free, since basepoint freeness is preserved bypullbacks and unions. The total space of a general pencil containing D′


is smooth, by the Bertini Theorems, and completing this pencil producesa divisor H in XK with a semistable model whose special fiber is D′. Inparticular, H is a smooth divisor in XK , equivalent to D′, with skeleton∆(D′), as required.

The analytification of XK is contractible, because X is smooth anddefined over the subfield C on which the t-adic norm is trivial. However,H1(Han,Z) and π1(Han) are isomorphic to Z/2Z. Thus, even thoughboth X and H are smooth, and H is ample in X, the natural maps

H1(XanK ,Z)→ H1(Han,Z) and π1(Han)→ π1(Xan

K )

are not isomorphisms.

5.5. Specialization from analytic points to closed algebraic sub-sets. Many semialgebraic subsets of analytifications that are of interestfor complex algebraic geometry are defined in terms of analytic pointsthat are near closed algebraic subsets. The notion of specialization cap-tures the rough idea of a point of Xan being close to a point, or moregenerally a Zariski closed subset of X.

Let x ∈ Xan be a point and let Z ⊂ X be a Zariski closed subset.

Definition 17. We say that x specializes into Z if it is represented by apoint x ∈ X(K) for a valued extension K|C with valuation ring R suchthat the inclusion ιx : SpecK → X extends to a morphism

ι : Spec (R)→ X,

which maps the closed point of SpecR into Z.

If X ⊂ Cn is affine, then the set of points specializing into Z is definedby the conditions |xi| ≤ 1 for 1 ≤ i ≤ n and |f | < 1 for f ∈ IZ . Giventhat the coordinate functions xi have norm less than or equal to 1, itsuffices to check that |fj| < 1 for some finite generating set {fj} of IZ ,so the set of points specializing into Z is semialgebraic.

Example 18. If x is a closed point of A1, then the set of points inthe analytification of A1 specializing to x is the open ray in Figure 3.1pointing from the central vertex toward x, together with x itself.

Example 19. If x is the closed point of A2 shown in Figure 3.4, thenthe entire open cone of monomial valuations specializes into x. The openrays pointing from η to ηX and ηX′ specialize into X and X ′, respectively,but not into x.

Example 20. If x is the node in Figure 5 then both open rays pointingfrom the central vertex toward x specialize to x, as does x itself.

24 SAM PAYNE

5.6. The analytic Milnor fiber. One prime example of a semialge-braic construction in nonarchimedean analytic geometry analogous to atopological construction in complex geometry is the analytic Milnor fiberof Nicaise and Sebag. Although their construction works more generally,we consider for simplicity the Milnor fiber of a single point in a hyper-surface, and choose coordinates so that this point is the origin 0 ∈ An.

Let X ⊂ Cn be the vanishing locus of a polynomial f , and assumethat X contains 0. Recall that the classical Milnor fiber of (X, 0) isthe intersection of the locus of points x ∈ Cn such that f(x) has fixedargument with a Euclidean sphere of radius ε � 1 centered at 0. See[Mil68] for further details.

We now define the nonarchimedean analytic Milnor fiber. Note thatthe polynomial f defines a morphism f an from the analytification of An

to the analytification of A1 = SpecC[y]. Fix some 0 < ε < 1, and let zbe the unique point of (A1)an such that |y|z = ε.

Definition 21. The analytic Milnor fiber of (X, 0) is the subset in theanalytification of An over C((t)) consisting of points x that specialize to0 such that f an(x) = z.

The condition that x specializes to 0 exactly means that x lies in theopen unit polydisc

D = {(x1, . . . , xn) | |xi| < 1 for 1 ≤ i ≤ n}.The analytic Milnor fiber is semialgebraic and definable over the normedextension C((t))|C in which |t| = ε; after base change to C((t)), it is justthe closed analytic subvariety of D defined by f = t. Therefore, its basechange to the algebraic closure of C((t)) carries an action of the absoluteGalois group, which induces an action on its `-adic etale cohomology.Nicaise and Sebag show that the `-adic etale cohomology of the analyticMilnor fiber, with the action of the procyclic generator of this Galoisgroup, is canonically identified with the `-adic singular cohomology ofthe classical Milnor fiber, with its monodromy action [NS07].

There are multiple advantages to the approach of Nicaise and Sebag.First, their definition makes sense in much greater generality, for vari-eties over an arbitrary field with the trivial norm, such as Fp, where onedoes not have the complex topology to work with. Furthermore, overC, their construction puts the Milnor fiber and its monodromy actionin a context (semialgebraic sets of smooth rigid varieties) where motivicintegration makes sense [LS03]. See also [HL11], which recasts the re-sulting interpretation of the motivic zeta function in the framework ofHrushovski and Kazhdan [HK06]. All of this work opens up possibilitiesfor a conceptual approach to the monodromy conjectures of Igusa [Igu75]and of Denef and Loeser [DL98].


5.7. The analytic link of a singularity. Another such constructionis the analytic link of a point x ∈ X(C). The classical link of x is atopological space Link(X(C), x) obtained by embedding an affine neigh-borhood of x in Cn and intersecting with a small sphere centered at x.It can be given a piecewise linear structure, by triangulating X(C) withx as a vertex and taking the link of this vertex in the resulting simplicialcomplex, and this triangulated space is well-defined up to piecewise lin-ear homeomorphism. Fundamental groups of links have been particularlyprominent in recent research activity; see the survey article [Kol13].

Definition 22. The analytic link Link(Xan, x) is the semialgebraic setin Xan r x consisting of points x′ that specialize to x.

Roughly speaking, this means that the points of the analytic link arethose close to x, but not x itself. The analytic link of an arbitrary closedalgebraic subset is defined similarly, and behaves like a deleted tubularneighborhood in classical complex geometry.

Advantages of the analytic link include the fact that its construction iscanonical, not depending on any choice of local embedding or triangula-tion, and that it carries the additional structure of an analytic space. Asfor the Milnor fibers discussed above, comparison theorems should givecanonical isomorphisms10

H∗et(Link(Xan, x),Q`) ' H∗(Link(X(C), x),Q`).

The weight zero piece of the etale cohomology of the link corresponds tothe singular cohomology of the underlying topological space, as in [Ber00,Nic11], so cohomological properties of singularities induce cohomologicalconditions on the topology of Link(Xan, x). For instance, if (X, x) is anisolated rational singularity, then Link(Xan, x) has the rational homologyof a point, and if (X, x) is an isolated Cohen-Macaulay singularity ofdimension n, then Link(Xan, x) has the rational homology of a wedge sumof spheres of dimension n−1. These conditions can also be interpreted interms of a log resolution; the dual complex of the exceptional divisor in alog resolution has the same homotopy type as the analytic link [Thu07].

Examples show that analytic links of rational singularities are not nec-essarily simply connected and may have torsion in their singular homol-ogy. See [Pay13, Example 8.1] and [KK11]. However there are strongernatural conditions on singularities that do imply contractibility of theanalytic link. For instance, toric singularities and finite quotient singu-larities have contractible analytic links [Ste06, KS12b].

10The analogous comparison theorem for Milnor fibers was proved by Nicaise andSebag [NS07], but to the best of our knowledge no such comparison for links hasappeared in the literature.

26 SAM PAYNE

De Fernex, Kollar, and Xu recently proved the strongest results in thisdirection, showing that analytic links are contractible for isolated logterminal singularities. Log terminal singularities are a subset of rationalsingularities that contain all toric and finite quotient singularities andappear naturally in birational geometry. The proof in [dFKX13] involvesthe study of dual complexes of exceptional divisors not only for log res-olutions, but for divisorial log terminal partial resolutions, and runninga carefully chosen minimal model program, while keeping track of howthese dual complexes transform at each step.

5.8. The analytic link at infinity. Analytic links of singularities area special case of analytic links at infinity. Again, we consider the ana-lytification of a variety X over the complex numbers with respect to thetrivial norm but now, instead of studying the singularity at a point x,we examine the failure of X to be compact.

Definition 23. The link at infinity Link∞(Xan) is the semialgebraic sub-set of Xan consisting of points that do not specialize to any point of X.

The link at infinity consists of points defined over valued extensions K|Cthat are not defined over the valuation ring R ⊂ K. If X is affine, thenx ∈ Link∞(Xan) if and only if |f |x > 1 for some polynomial f ∈ C[X]. Bythe triangle inequality, it is enough to consider f in some finite generatingset, such as the coordinate functions for some embedding X ⊂ Cn.

Example 24. Suppose X is compact. Then Link∞(Xan) is empty and,for any point x ∈ X, the space Link(Xan, x) coincides with Link∞(Xrx).

Example 25. Suppose X is the nodal curve with three punctures dis-cussed in Example 13. Then Link∞(Xan) consists of the three open raysin Figure 5 pointing at the three punctures.

Remark 26. The link at infinity is also characterized as the link ofthe boundary in any compactification of X. In other words, if X is acompactification of X with boundary ∂X = X r X the Link∞(Xan) isthe set of points of Xan that specialize to a point in ∂X.

The link at infinity is related to compactifications in much the same waythat the link of an isolated singularity is related to resolutions. SupposeX is smooth and X is a smooth compactification whose boundary

∂X = X rX

is a divisor with simple normal crossings. Then Thuillier’s constructionsgives a canonical homotopy equivalence from Link∞(X) to the dual com-plex ∆(∂X) [Thu07]. One can then use excision exact sequences and


Poincare duality to identify the rational cohomology of Link∞(Xan) withthe top graded piece of the weight filtration on the cohomology of X(C),

H i(Link∞(Xan),Q) ∼= GrW2 dimXH2 dimX−i(X(C),Q).

Such constructions with dual complexes of boundary divisors in smoothcompactifications are surveyed in [Pay13], but the main ideas were intro-duced and studied much earlier by Danilov [Dan75]. This framework canbe extended to toroidal compactifications of smooth Deligne-Mumfordstacks, and this generalization has been applied to compute top weightcohomology groups for many moduli spaces of stable curves with markedpoints, includingM1,n for all n, using an interpretation of the dual com-plex of the boundary of the Deligne-Mumford compactification as a mod-uli space for stable tropical curves [ACP12, CGP14].

Acknowledgments. I thank P. Achinger, M. Baker, D. Cartwright,L. Fantini, S. Grushevsky, J. Huh, A. Kontorovich, J. Nicaise, D. Ran-ganathan, D. Speyer, W. Veys, and the referee for helpful comments andimportant corrections to earlier versions of this article, and am gratefulto D. Ranganathan also for assistance in preparing the figures.

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